CONVEX COMBINATIONS OF ANALYTIC FUNCTIONS

Transcription

1 rnat. J. Math. & Math. S. Vl. 6 N. (983) ON THE RADUS OF UNVALENCE OF CONVEX COMBNATONS OF ANALYTC FUNCTONS KHALDA. NOOR, FATMA M. ALOBOUD and NAEELA ALDHAN Mathematcs Department Scence Cllege f Educatn fr Grls Mala, Stteen Rad Ryadh, SAUD ARABA (Receved July, 98 and n revsed frm February, 983) ABSTRACT. We cnsder fr a >, the cnvex cmbnatns f() (la)f() + af (), where F belngs t dfferent subclasses f unvalent functns and fnd the radus fr whch f s n the same class. KEY WORDS AND PHRASES. Unvalent funtns, alphaquascnvex, stake, clsetcnvex funct, nvex cmbnns. 98 MATHEMATCS SUBJECT CLASSFCATON CODES. Prmary 3A3, Secndary 3A34.. NTRODUCTON. Let S, K, S* and C dente the classes f analytc functns n the unt dsc E {: l < } whch are respectvely unvalent, clsetcnvex, starlke, and cnvex. n [,], a new subclass C* f unvalent functns was ntrduced and studed. A functn f, analytc n E, belngs t C* f and nly f there exsts a cnvex functn g such that fr g E, Re (f ()) > O. (.) Tle functns n C* are called quascnvex and C C* K S. t s shwn [] that f g C* f and nly f f g K. Recently the functns called aquascnvex have been defned and ther prpertes studed n [3]. A functn f, analytc n E, s sad t be squascnvex f and nly f there exsts a cnvex functn g such that, fr a real and pstve f () Re{(l a) () + a (f (Z)) > O, (.)

2 336 K.. NOOR, C. M. ALOBOUD and N. ALDHAN t has been shwn [3] that F s squaslcnvex f and nly f f wth f() ( e)f() + F () s clsetcnvex. (.3) All squaslcnvex functns are clsetcnvex.. MAN RESULTS. We shall nw study the mappng prpertes f f: f() ( )F() + F (), e >, when F belngs t dfferent subclasses f unvalent functns. THEOREM.. Let F S* and >. The functn F() ( )F() + F () (.) s starlke n l < r, where r + + (.) Ths result s sharp. PROOF. We can wrte (.) as f() (Z F()), and frm ths t fllws that Then c ).F () {(( [ F() F() {( { e f()d + where Re h() >, snce F e S*. Frm (.4), we have Z f() ) C Dfferentatng bth sdes f (.5), we btan ( ) f() + f () ) C Thus () h(),+ {h () O f() f()d f()d + f())/( f()d)} f()}/{ f()d} h(), f()d h() C f()d. f(z) f() =h () f()d+h() e f()d}/{ O f()}. (.3) (.4) (.5)

3 UNVALENCE OF CONVEX COMBNATONS OF ANALYTC FUNCTONS 337 Nw, usng the wellknwn result [4], lh () < {Re h()}/(l r), l r, we have Re Frm (.) and (.3), we have f() f ()> Re h(){l F()) f( ( f()d ( F()) frm whch t fllws that l{f()/ f()d }" (.6) r f() { c F () + ) F()} ( c F() F () F) + ) h() + ( ), r (.7) f()d} > Re{h() + )} > () + + r Re f () r > Re h(){l + r )} f() + r c (_ )r Re h ){(_.._. le 4r )r)}/{( r)( + r)} (.8) f () {C(( )))}/(l ) ( )F () + OF (), (.9) Usng (.7), we have frm (.6) The rght hand sde f (.8) s pstve fr r < r where r s gven by (.). Ths result s sharp as can be seen by where F () ( ) S* RE}L%RK.. Let f e C, then f, gven by (.), s cnvex fr l < r where r s gven by (.). The prf fllws n the same lnes as n Therem.. See als [5] and [6]. REMARK.. n [6], Nklaeva and Repnna treated the same prblem, wth a dfferent ntatn, fr the cnvex and starlke functns f rder 8. Therem. fllws frm ther result when we take 8 fr < <. On the ther hand, ur prf f Therem.s much smpler and the result hlds fr all >.

4 338 K.. NOOR, F. M. ALOBOUD and N. ALDHAN THEOREM.. Let F e K and f() ( e)f() + F (), > O. Then f s clsetcnvex n l < r r s gven by ( ). The functn f n ( 9) shws that ths result s sharp. PROOF. Snce F E K, there exsts a G e S* such that, fr E E, Re F () G() Nw let g() ( )G() + G (). Then by Therem. g s starlke fr l <r O r s defned by (.). Usng the same technque f Therem we can easly shw O that Re f ( > fr l < r g() REMARK.3. Fr a / ths result has been prved n [7]. As an easy cnsequence f (.3) and Therem., we have the fllwng. COROLLARY.. Let F e K and f() ( e)f() + af (), a > O. Then F s e quascnvex n l < r Ths means that the radus f aquascnvexty fr clse O tcnvex functns s gven by (.). THEOREM.3. Let F g C* and a >. Let f() ( a)f() + af (). Then f s n C*, fr l < r, r s gven by (.). PROOF. Snce F C*, there exsts a G e C such that fr e E, Re (F ()) G () Nw let g() ( )G() + G (), then g s cnvex n l < r O We can wrte f() ( a)f() + af () F()) >. >, and g() ( e)g() + eg () G()) Thus Nw (f () ) (( ( F()) ) ) ((()F() E E ((( F()) ) ) )/( (( O()) ) ) + F ()) ) F () + ()) F" (.). (Z Z Let F () H(), then frm (.), we have F () + F"())) ( ( F ()) ) (f ()) e a a ( H()) ) / ( g G()) ) () Snce frm Therem., the functn (l)tt() + tt () t()) be. lngs t K wth respect t a cnvex functn g: g() (le)g() + eg () n

5 UNVALENCE OF CONVEX COMBNATONS OF ANALYTC FUNCTONS 339 l < r s f s n C* fr l < r where r s gven by (.). REMARK.4. Fr F e C* and /, Therem.3 has been prved n []. We nw deal wth a generaled frm f (.) by takng g t be starlke and prve the fllwng. THEOREM.4. Let F be analytc n E and let fr e E, Re (F ()) G () Let f() (l)f() + F () and g() (l)g() + G (), wth e >. Re (.f ()) > fr l < rl, where >, GeS*. Then r Fr e /, the prblem has been slved n [8]. PROOF. Frm (.3), we can wrte F() C f()d [ F (.) (() j f()d + a f()) Thus (F ()) G () a f () d) (ef () () f f ()d) a g ()d h (), (.) where Re h() >, E. Frm (.), we wrte f, () () f ()d h() g ()d Dfferentatng bth sdes, and smplfyng, we btan Usng lh ()] (f ()) =h() + < Re h() (.) gves r h ()( J g () d) (.)

6 34 K.. NOOR, F. M. ALOBOUD and N. ALDHAN Re (f ())_ g C) > Re h()[l r a g () d) / ( g ()d) ] (.3) Nw E (g ())/( g ()d) (/)G () + G"() G () () + (G ()) G () (.4) Snce G e S*, s (G ()) G () l4r+r (.5) Frm (.3), (.4) and (.5), we btan (f ()) Re > Re r(l r) h()[l r 4r )r 6r Re h() ( )r 4er ( )r and ths pstve fr l < rl, where r 3c + /9a + a ACKNOWLEDGEMENT. The authrs are grateful fr the referee s helpful cmments and suggestns n the earler versn f ths paper. n partcular, the reference t Nklaeva and Repnna was kndly suppled by hm. REFERENCES. NOOR, K. NAYAT. On a subclass f clsetcnvex functns, Cmm. Math. Unv. St. Paul 9 (98), 58.. NOOR, K. NAYAT and THOMAS, D.K. On quascnvex unvalent functns, nter. J. Math. and Math. Sc. 3 (98), NOOR, K. NAYAT and ALOBOUD, F.M. Alphaquascnvex functns, t appear. 4. LBERA, R.J. Sme radus f cnvexty prblems, Duke Math. J. 3 (964), CAMPBELL, D.M. A survey f prblems f the cnvex cmbnatn f unvalent functns, Rckey Munt J. Math. 5 (975), NKOLAEVA, R.V. and REPNNA, L.G. A certan generalatn f therems due t Lvngstn, (Russan), Ukran Mat. Z. 4 (97), 6873, MR45 # LVNGSTON, A.E. On the radus f unvalence f certan analytc functns, Prc. Amer. Math. Sc. 7 (966), NOOR, K NAYAT and ALDHAN, N. A subclass f clsetcnvex functns, t appear.